NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

96 GUILLARMOU and TZOU
enough, we have the estimate
C
ɛ u2 L2
(M0) + C′ − Im(〈∂τ u, u〉L2 (∂M0)) + 1
h
∂M0
|u| 2 ∂νϕɛdvg
≤e −ϕɛ/h¯∂e ϕɛ/h 2
uL2 (M0) , (4)
where ∂ν and ∂τ denote, respectively, the exterior pointing normal vector fields and
its rotation by an angle +π/2.
Proof
We use complex coordinates z = x +iy in the chart , where u is supported. Since the
Lebesgue measure dxdy is bounded below and above by dvg, g is conformal to |dz| 2 ,
and the boundary terms in (4) depend only on the conformal class, it then suffices to
prove the estimates with respect to dxdy and the Euclidean metric. We thus integrate
by parts with respect to dxdy and we have
4e −ϕɛ/h¯∂e ϕɛ/h
2
u = ∂x + i∂yϕɛ
u + i∂y +
h
∂xϕɛ
u
h
2
= ∂x + i∂yϕɛ
u
h
2
+ i∂y + ∂xϕɛ
u
h
2
+ 2
ϕɛ|u|
h
2 − 1
2 ∂xϕɛ.∂x|u| 2 − 1
2
∂yϕɛ.∂y|u|
2
+ 2
∂νϕɛ|u|
h ∂M0
2
− 2 ∂xRe(u).∂yIm(u)
M0
− ∂xIm(u).∂yRe(u)
(5)
= ∂x + i∂yϕɛ
u
h
2
+ i∂y + ∂xϕɛ
u
h
2
+ 1
ϕɛ|u|
h
2
+ 1
∂νϕɛ|u|
h ∂M0
2
+ 2 ∂τ Re(u).Im(u),
∂M0
where := −(∂ 2 x + ∂2 y ), where ∂ν is the exterior pointing normal vector field to the
boundary, and where ∂τ is the tangent vector field to the boundary (i.e., ∂ν rotated
with an angle π/2) for the Euclidean metric |dz| 2 .Then〈uϕɛ,u〉=(h/ɛ)(|dϕ0| 2 +
|dϕ1| 2 +···+|dϕN| 2 )|u| 2 ,sinceϕj are harmonic, so the proof follows from the fact
that |dϕ0| 2 +|dϕ1| 2 +···+|dϕN| 2 is uniformly bounded away from zero.
Themainsteptogofrom(4) to(3) is the following lemma, whose proof is a slight
modification of the one in [19, Proposition 5.3].